The reason that photographers use rangefinders for street photography is because HCB did it. But HCB could have taken wonderful photographs with a Kodak Brownie if it was all he had available. But then again, if he had, we would probably be spending thousands of dollars for Brownie cameras and lenses...
Rangefinders are simpler optically than SLRs, but focus more slowly (depending on how much practice you have had), and less precisely than SLR cameras. This difference in focus precision probably outweighs the benefit of having no mirror vibration. I tinker with lots of cameras, and more than half of the rangefinder cameras I've come across are in need of rangefinder adjustments to some degree.
Also, though rangefinders can used wonderful lenses, this number of lenses is small, as cameras like the Leica M series have focusing lines for 4 lenses. For different lenses you may have to use an accessory viewfinder. With an SLR, you can use any lens which will fit the mount, and see what the lens sees when you look through the viewfinder.
As for myself, I shoot a rangefinder most of the time. Not because it's better or worse than an SLR, but simply because I like to.